Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Extending bilinear elliptic curve pairings to multilinear maps is a long-standing open problem. The first plausible construction of such multilinear maps has recently been described by Garg, Gentry and Halevi, based on ideal lattices. In this paper we describe a different construction that works over the integers instead of ideal lattices, similar to the DGHV fully homomorphic encryption scheme. We also describe a different technique for proving the full randomization of encodings: instead of Gaussian linear sums, we apply the classical leftover hash lemma over a quotient lattice. We show that our construction is relatively practical: for reasonable security parameters a one-round 7-party Diffie-Hellman key exchange requires less than 40 seconds per party. Moreover, in contrast with previous work, multilinear analogues of useful, base group assumptions like DLIN appear to hold in our setting.

/pdf/practical-multilinear-maps-over-the-integers-4ie6ysydhv.pdf

Practical Multilinear Maps over the Integers

Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.

/pdf/modular-forms-a-computational-approach-8izt8ndhca.pdf

Modular forms, a computational approach

According to the Oxford English Dictionary, the word topology is derived of topos (tìpo ) meaning place, and -logy (logia), a variant of the verb légein, meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer. The field of geometry studies intrinsic properties that are invariant under rigid motion, such as the curvature of a surface. In contrast, topology studies invariants under continuous deformations. The larger set of transformations enables topology to extract more qualitative information about a space, such as the number of connected components or tunnels. Computational topology has theoretical and practical goals. Theoretically, we look at the tractability and complexity of each problem, as well as the design of efficient data structures and algorithms. Practically, we are interested in heuristics and fast software for solving problems that arise in diverse disciplines. Our input is often a finite discrete set of noisy samples from some underlying space. This type of input has renewed interest in combinatorial and algebraic topology, areas that had been overshadowed by point set topology in the last one hundred years. Computational topology developed in response to topological impediments emerging from within geometric problems. In computer graphics, researchers encountered the problem of connectivity in reconstructing watertight surfaces from point sets. Often, their heuristics resulted in surfaces with extraneous holes or tunnels that had to be detected and removed before proper geometric processing was feasible. Researchers in computational geometry provided guaranteed surface reconstruction algorithms that depended on sampling conditions on the input. These results required concepts from topology, provoking an interest in the subject. Topological problems also arise naturally in areas that do not deal directly with geometry. In robotics, researchers need to understand the connectivity of the configuration space of a robot for computing optimal trajectories that minimize resource consumption. In biology, the thermodynamics hypothesis states that proteins fold to their native states along such optimal trajectories. In sensor networks, determining coverage without localizing the sensors requires deriving global information from local connections. Once again the question of connectivity arises, steering us toward a topological understanding of the problem. Like topology, computational topology is a large and diverse area. The aim of this chapter is not to be comprehensive, but to describe the fundamental concepts, methods, and structures that permeate the field. We begin with our objects of study, topological spaces and their combinatorial representations, in Section 2. We study these spaces through topological invariants, which we formalize in Section 3, classifying all surfaces, and introducing both combinatorial and algebraic invariants in the process. Sections 4 and 5 focus on homology and persistent homology, algebraic invariants that derive their popularity from their computability. Geometry and topology are intrinsically entangled, as revealed by Morse theory through additional

Computational topology

Computing canonical forms of matrices over rings is a classical math¬ ematical problem with many applications to computational linear alge¬ bra. These forms include the Frobenius form over a field, the Hermite form over a principal ideal domain and the Howell and Smith form over a principal ideal ring. Generic algorithms are presented for computing each of these forms together with associated unimodular transformation matrices. The algorithms are analysed, with respect to the worst case, in terms of number of required operations from the ring. All algorithms are deterministic. For a square input matrix, the algorithms recover each of these forms in about the same number of operations as required for matrix multiplication. Special emphasis is placed on the efficient computation of transforms for the Hermite and Smith form in the case of rectangular input matrices. Here we analyse the running time of our algorithms in terms of three parameters: the row dimension, the column dimension and the number of nonzero rows in the output matrix. The generic algorithms are applied to the problem of computing the Hermite and Smith form of an integer matrix. Here the complexity anal¬ ysis is in terms of number of bit operations. Some additional techniques are developed to avoid intermediate expression swell. New algorithms are demonstrated to construct transformation matrices which have good bounds on the size of entries. These algorithms recover transforms in essentially the same time as required by our algorithms to compute only the form itself. Kurzfassung Kanonischen Formen von Matrizen über Ringen zu berechnen, ist ein klassisches mathematisches Problem mit vielen Anwendungen zur konstruktiven linearen Algebra. Diese Formen umfassen die Frobenius Form über einem Körper und die Hermite-, Howellund Smith-Form über einem Hauptidealring. Wir studieren die Berechnung dieser Formen aus der Sicht von sequentiellen deterministischen Komplexitätsschranken im schlimmsten Fall. Wir präsentieren Algorithmen für das Berechnen aller dieser Formen sowie der dazugehörigen unimodularen Transforma¬ tionsmatrizen samt Analyse der Anzahl benötigten Ringoperationen. Die Howell-, HermiteSmithund Frobenius-Form einer quadratischen Matrix kann mit ungefähr gleich vielen Operationen wie die Matrixmul¬ tiplikation berechnet werden. Ein Schwerpunkt liegt hier bei der effizienten Berechnung der Hermiteund Smith-Form sowie der dazugehörigen Transformationsmatrizen im Falle einer nichtquadratischen Eingabematrix. In diesem Fall analysieren wir die Laufzeit unserer Algorithmen abhänhig von drei Parametern: die Anzahl der Zeilen, die Anzahl der Spalten und die Anzahl der Zeilen in der berechneten Form, die mindestens ein Element ungleich Null enthal¬ ten. Die generische Algorithmen werden auf das Problem des Aufsteilens der Hermiteund Smith-Form einer ganzzahligen Matrix angewendet. Hier wird die Komplizität des Verfahren in der Anzahl der benötigten Bitoperationen ausgedrückt. Einige zusätzliche Techniken wurden en¬ twickelt, um das übermässige Wachsen von Zwischenergebnissen zu ver¬ meiden. Neue Verfahren zur Konstruktion von Transformationsmatrizen für die Hermiteund Smith-Form einer ganzzahligen Matrix wurden en¬ twickelt. Ziel der Bemühungen bei der Entwicklung dieser Verfahren war im Wesentlichen das erreichen der gleichen obere Schranke für die Laufzeit, die unsere Algorithmen benötigen, um nur die Form selbst zu berechnen.

Algorithms for matrix canonical forms

We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d For the case of matrices over Z d we present an algorithm that computes the Smith form S of an A Z n m d in only O n m operations from Z d Here is the ex ponent for matrix multiplication over rings two n n matrices over a ring R can be multiplied in O n opera tions from R We apply our algorithm for matrices over Z d to get an algorithm for computing the Smith form S of an A Z n m in O n m M n log jjAjj bit opera tions where jjAjj max jAi jj and M t bounds the cost of multiplying two dte bit integers These complexity re sults improve signi cantly on the complexity of previously best known Smith form algorithms both deterministic and probabilistic which guarantee correctness

Near optimal algorithms for computing Smith normal forms of integer matrices

On lattice reduction for polynomial matrices

This paper presents a new algorithm for computing the Hermite normal form H of an A Z n m of rank m to gether with a unimodular pre multiplier matrix U such that UA H Our algorithm requires O m nM m log jjAjj bit operations to produce both H and a candidate for U Here jjAjj maxij jAijj M t bit operations are su cient to multiply two dte bit integers and is the exponent for matrix multiplication over rings two m m matrices over a ring R can be multiplied in O m ring operations from R The previously fastest algorithm of Hafner McCurley re quires O m nM m log jjAjj bit operations to produce H but does not produce a candidate for U Previous methods require on the order of O n M m log jjAjj bit operations to produce a candidate for U our algorithm improves on this signi cantly in both a theoretical and practical sense

/pdf/asymptotically-fast-computation-of-hermite-normal-forms-of-1cghq2325r.pdf

Asymptotically fast computation of Hermite normal forms of integer matrices

Reductions to polynomial matrix multiplication are given for some classical problems involving a nonsingular input matrix over the ring of univariate polynomials with coefficients from a field. High-order lifting is used to compute the determinant, the Smith form, and a rational system solution with about the same number of field operations as required to multiply together two matrices having the same dimension and degree as the input matrix. Integrality certification is used to verify correctness of the output. The algorithms are space efficient.

Arne Storjohann

Papers

Algorithms for matrix canonical forms

Near optimal algorithms for computing Smith normal forms of integer matrices

On lattice reduction for polynomial matrices

Asymptotically fast computation of Hermite normal forms of integer matrices

High-order lifting and integrality certification